The Mathematical Mystery of Nebulous Negatives
Multiplication of negatives is generally thrown about as a rule, but there's more to it when we look at Brahmagupta's Treatise
The introduction of negative numbers start somewhere between the 4th and 6th grades. These are usually taught using a number line and it works fairly well for addition, and to an extent for subtraction.
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However, as soon as we get to multiplication involving negatives, this starts becoming an issue, especially a negative multiplied by a negative.
Even if you ask your Math teacher why -2 x -2 results in +4, usually they respond with, “It’s a rule!”
Have you ever wondered where that rule came from?
Before that, a bit of Math history. In the West, especially in Europe, it started being defined in the 15th Century CE. An English Mathematician invented the number line in the 17th Century CE and they actually started getting used in the 19th Century CE. In fact famous British Mathematicain and Logician, Augustus De Morgan (1806 - 1871), one of the people who influenced computing, has been known to have called negative numbers as “evil”.
De Morgan was one of the trio of Babbage, De Morgan and Boole who are considered the people who laid the foundations for computational sciences, along with their friend and acquaintance Henry Colebrooke, a translator of several Sanskrit works.
Did you know that Europe was using Roman Numerals up until the 14th Century with Britain using it up to the 15th Century, and it was still being used for bookkeeping in some places all the way to the 18th Century CE?
So, obviously they wouldn’t have known negative numbers well enough to do any arithmentic with them.
Rewind to around 6th Century CE, India. Well known mathematician Brahmagupta not only had rules for arithmetic, but rules for both positive and negative numbers that far back.
Now, do keep in mind, while the Indians were extremely advanced in Mathematics, it essentially was simply a tool for serving a purpose. So, you’ll find chapters on things like Arithmetic in treatises like Brahmasphūṭa Siddhānta which is essentially an astronomical treatise, or you’ll find practical geometry in the Śulba Sūtras which are a part of the Vedic corpus.
We saw the algorithmic square rooting in the Gaṇitadīpikā section of Āryabhaṭiyam here →
We find the rules for Arithmetic of positive and negative numbers in the Kuṭṭakādhyaya which is chapter 18, and not in Gaṇitādhyaya which is chapter 12. At first glance for folks who do not know what is Kuṭṭaka, this doesn’t make sense. However, Kuṭṭaka is about algebraic expressions on indeterminate equations and having rules for positive and negative numbers makes sense just because of that.
Anyway, it starts at shloka number 31 which states the rules for addition of positive and negative numbers and zeros. The terminology for positive numbers is धनम् (dhanam) and for negative numbers is ऋणम् (Ṛṇam) which makes sense as dhanam is also used for wealth and ṛṇam is also used for debt.
धनयोर्धनम् ऋणमृणयोः धनर्णयोरन्तरं समैक्यं खम् ।
ऋणमैक्यम् च धनमृणधनशून्ययोः शून्ययोः शून्यम् ॥
We’ll now break this down as understanding some of these will become important to understand multiplication. Before we get into the nitty gritty, a slight digression as Sanskrit has this dual form for word forms. So, here’s the table for Dhana
What is important to understand is the षष्ठी (Sixth) form which is the Genetive case (of, belongs to etc). The dual form is धनयोः which essentially translates to “of two positives” and similarly ऋणयोः is “of two negatives”.
Okay, so the first part of the shloka for addition is:
धनयोर्धनम् = धनयोः धनं (Of two positives is a positive)
Or Positive + Positive = Positive
ऋणमृणयोः = ऋणम् ऋणयोः = ऋणयोः ऋणम् (Of two negatives is a negative)
Or Negative + Negative = Negative
The next is a mix
धनर्णयोरन्तरं = धनर्णयोः अन्तरं = धन ऋणयोः अन्तरं = धनस्य ऋणस्य अन्तरं
This can be a little confusing, however the योः represents two and this is a common practice in Sanskrit.
This basically says that the result depends on the difference (As we know the sign will take the one of the larger magnitude).
Or Positive + Negative = Positive/Negative
This next one is the key to a lot of things
समैक्यं खम् = सम् ऐक्यं खम्
This translates to when the positive and negative are of equal magnitude, it results in zero. We’ll actually be using this particular formula of 0 to understand negative multiplication.
The next line of the shloka is about addition of zeros to a positive or negative number
ऋणमैक्यम् च धनमृणधनशून्ययोः which Brahmagupta has explained in his own commentary
Positive + Zero = Positive
Negative + Zero = Negative
And of course, the last one is
शून्ययोः शून्यम्
Or Zero + Zero = Zero
Similarly, to the rules for saṅkalana (addition) of positive numbers, negative numbers and zero, there are rules for vyavakalana (subtraction), guṇana (multiplication) and haraṇa (division).
Now, the rule for negative multiplication is
धनमृणयोः = धनं ऋणयोः = ऋणयोः धनं
Or Negative * Negative = Positive
Now, we’ll start working on trying to understand it.
Before we do that, we need to understand multiplication as repeated addition. Now, for the longest time, it was stated that m times n is m added to itself n times. This definition is actually incorrect.
For example 1 x 1 = 1 times 1 = 1 added to itself 1 time = 1 + 1 = 2.
So, first we have to correct this and rephrase this as
m times n is m added to 0 n times
One we do this, 1 x 1 = 0 + 1 = 1
And 2 x 2 = 0 + 2 + 2 = 4
Similarly 2 x -2 = 0 + (-2) + (-2) = -4 and these are now consistent with Brahmagupta’s rules for addition as well as multiplication.
Now, we’re going to do something I actually learnt from a Math researcher named Jonathan Crabtree. He used Brahmagupta’s definition of zero as समैक्यं खम् as in opposites of equal magnitudes are zero, as in
(-x) + x = 0
-4 + 4 = 0
-9 + 9 = 0
-9384 +9384 = 0
So, let us take multiplication of -2 x -2 which can be confusing.
2 x -2 was 0 + (-2) + (-2) so to make it -2 x -2 lets multiply the first part by -1 as in
(-1 x 2) x -2 which is -1 x (2 x -2) which is -1 x (0 + -2 + -2)
which results in 0 - (-2) - (-2)
But, we are faced with the prospect of removing a negative from zero. How do you do that, and better yet, how do you show that to a child?
Let’s take some balls. The red balls represent positive quantities and the other is say negative. So, the number of balls is the number.
The above is 2 + (-2) and it is 0.
Now, let’s multiply. 2 x 2 and -2 x 2
So far, it’s a piece of cake. You can easily see the results of the 4 positives and 4 negatives. Now, for -2 x -2, as we had discussed above, we need to remove 2 negatives from 0.
We can replace the zero using समैक्यं खम् as follows
As you can see, the negatives cancel each other out, and you’re left with 4 positives.
The Brahmasphūṭa Siddhānta is the first available text we have available today that discusses the mathematics of zero (śūnya-parikarma) along with the rules for positives and negatives.
The rules for arithmetic operations on positive and negative numbers are given in the 18th chapter or the Kuṭṭakādhyaya (algebra of indeterminate forms). There’s also arithmetic in the 12th chapter called gaṇitādhaya.
This is a book on astronomy and has a lot of advanced concepts such as the calculations of eclipses and other phenomenon, and is actually quite a large work that spans multiple volumes.
To find mathematical content in this that can be used for teaching school children is rather interesting, especially things that were mistranslated or not completely understood as they travelled west.
We’re lucky to still have these marvels still existing despite the bigoted library burnings of the Islamic raiders like Bakhtiyar Khilji. Eye-witness and scribe accounts state that these books burnt somewhere from 3 to 6 months.
Other notable universities which got destroyed that were known to have vast libraries included the neighbouring Vikramshila, Vallabhi, Nagarjuna etc which does point to a culture of libraries. The destruction was a big setback to a knowledge based culture and civilization.
The vast amount of manuscripts available today in Sanskrit as well as several references they have also points to the existence of a culture of documentation, besides the oral tradition which helped keep some of the vedic corpus alive.
During my visit to a gurukula that teaches vaijnanik (scientific) subjects like Gaṇitam, Nyāya, Khagola, Jyotisha, Vyākaraṇa etc, I found that despite having the books, the oral transmission culture is there in the teaching prakriya (pedagogy) which always starts with kanṭhasthikaraṇam (memorization) of the content first. Then there is a process which is still alive today and it’s even the same with Math. The students knew the entire Līlāvati by heart and also how to use it.
There’s a lot of merit and a lot of knowledge in many of these works, but it hasn’t yet been pulled out for understanding as Mathematics due to people not knowing this incredible legacy and the availability of such treatises.
References
Brahmasphūṭa Siddhanta → https://archive.org/details/Brahmasphutasiddhanta/page/n17/mode/2up
Jonathan Crabtree’s Paper on Brahmagupta → https://www.jonathancrabtree.com/mathematics/brahmaguptas-18-maths-sutras-of-symmetry/
16 ancient universities of India → https://www.myindiamyglory.com/2019/02/14/15-ancient-universities-of-india-from-3600-plus-years-ago/
The Imperishable Seed by Bhaskar Kamble → https://garudabooks.com/the-imperishable-seed
Meru Prastaar by Chandrahas Halai → https://garudabooks.com/meru-prastaar